[Paper Review] A New Algorithm for Inverting General Cyclic Heptadiagonal Matrices Recursively

A. A. Karawia|arXiv (Cornell University)|Nov 10, 2010

Scientific Research and Discoveries12 references3 citations

TL;DR

This paper presents a symbolic, recursive algorithm—CHINV—for inverting general cyclic heptadiagonal matrices without restrictive assumptions on matrix elements. It uses Doolittle LU factorization, parallel computation, and recursive relations to compute the inverse column-by-column, with the final inverse derived via symbolic simplification and substitution of a parameter to handle potential zero denominators.

ABSTRACT

In this paper, we describe a reliable symbolic computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel computing along with recursion. The algorithm is implementable to the Computer Algebra System(CAS) such as MAPLE, Matlab and Mathematica . An example is presented for the sake of illustration.

Motivation & Objective

• To develop a reliable, symbolic algorithm for inverting general cyclic heptadiagonal matrices without restrictive assumptions on matrix elements.
• To extend existing algorithms for pentadiagonal and heptadiagonal matrices to a fully general cyclic heptadiagonal case.
• To ensure numerical robustness by using symbolic computation to avoid division-by-zero failures.
• To provide an efficient, recursive method for solving linear systems involving cyclic heptadiagonal matrices.
• To generalize prior recursive algorithms for pentadiagonal matrices to the heptadiagonal case.

Proposed method

• Employs Doolittle LU factorization to decompose the cyclic heptadiagonal matrix H into L and U matrices with symbolic expressions for elements.
• Computes the last four columns of the inverse matrix H⁻¹ using recursive relations derived from LU factorization and symbolic simplification.
• Uses a symbolic parameter t to handle potential zero denominators in the inverse computation, later substituting t = 0 to obtain the final result.
• Applies recursive column relations (Eqs. 2.39–2.40) to compute the remaining (n−5) columns of H⁻¹ from the known last four columns.
• Implements the algorithm in a Computer Algebra System (CAS) such as MAPLE, MATLAB, or MATHEMATICA for symbolic execution.
• Validates nonsingularity via a DETCHEPTA sub-algorithm before computing the inverse.

Experimental results

Research questions

• RQ1 Can a general cyclic heptadiagonal matrix be inverted without imposing restrictive conditions on its elements?
• RQ2 How can symbolic computation be used to ensure numerical reliability in matrix inversion algorithms?
• RQ3 What recursive structure allows efficient computation of the inverse of a cyclic heptadiagonal matrix beyond the standard pentadiagonal case?
• RQ4 How can the inverse be computed column-by-column using LU factorization and recursive relations?
• RQ5 What symbolic parameterization strategy prevents division-by-zero failures during inversion?

Key findings

• The CHINV algorithm successfully computes the inverse of a general cyclic heptadiagonal matrix of order n ≥ 8 without restrictive assumptions on matrix elements.
• The algorithm is robust and failsafe due to symbolic parameterization with t, which is later set to zero to yield the final inverse.
• The inverse is computed column-by-column: the last four columns are derived via recursive relations (Eqs. 2.15–2.38), and the remaining columns via Eqs. 2.39–2.40.
• The method generalizes prior recursive algorithms for pentadiagonal matrices, extending their applicability to heptadiagonal systems.
• In the illustrative example with n=10, the algorithm correctly recovers the solution vector X = [1,2,3,4,5,6,7,8,9,10] for the given linear system.
• The algorithm is implementable in CAS environments like MAPLE, MATLAB, and MATHEMATICA, enabling exact symbolic computation.